First Cells, Then Species, Now the Web

By GEORGE JOHNSON

Published: December 26, 2000

As the Internet continues to proliferate, it has become natural to think of it biologically -- as a flourishing ecosystem of computers or a sprawling brain of Pentium-powered neurons. However you mix and match metaphors, it is hard to escape the eerie feeling that an alien presence has fallen to earth, confronting scientists with something new to prod and understand.

The result has been an eruption of papers scrutinizing this artificial network and concluding, to many people's surprise, that it may be designed according to the same rules that nature uses to spin webs of its own. The networks of molecules in a cell, of species in an ecosystem, and of people in a social group may be woven on the same mathematical loom as the Internet and the World Wide Web.

''We are getting to understand the architecture of complexity,'' said Dr. Albert-Laszlo Barabasi, a physicist at the University of Notre Dame in Indiana whose research group has recently published papers comparing such seemingly diverse systems as the Internet and the metabolic networks of life-sustaining chemical reactions inside cells. The similarities between these and other complex systems are so striking, he said, ''it's as if the same person would have designed them.''

At the Polytechnic University of Catalonia in Barcelona, Dr. Ricard V. Sole and Jose M. Montoya, theoretical biologists in the Complex Systems Research Group, have recently found the same kind of patterns by studying computer models of three ecosystems: a freshwater lake, an estuary and a woods. ''These results suggest that nature has some universal organizational principles that might finally allow us to formulate a general theory of complex systems,'' said Dr. Sole, who also works at the Santa Fe Institute in New Mexico.

In the past, scientists treated networks as though they were strung together at random, giving rise to a homogeneous web in which nodes tended to have roughly the same number of links. ''Our work illustrates that in fact the real networks are far from being random,'' Dr. Barabasi said. ''They display a high degree of order and universality that has been rather unexpected by any accounts.''

As they come together, many networks seem to organize themselves so that most nodes have very few links, and a tiny number of nodes, called hubs, have many links. The pattern can be described by what scientists call a power law. To calculate the probability that a node will have a certain number of links, you raise that number to some power, like 2 or 3, and then take the inverse.

Suppose, for example, that you have a network with 100,000 nodes that obeys a power law of 2. To find out how many nodes have three links, you raise 3 to the second power, which is 9, and then take the inverse. Thus one-ninth of the nodes, or about 11,111, will be triple linked. How many will have 100 links? Raise 100 to the second power, and take the inverse: one ten-thousandth of the 100,000 nodes -- a total of 10 -- will be so richly connected. As the number of connections rises, the probability rapidly falls.

This kind of structure may help explain why networks ranging from metabolisms to ecosystems to the Internet are generally very stable and resilient, yet prone to occasional catastrophic collapse. Since most nodes (molecules, species, computer servers) are sparsely connected, little depends on them: a large fraction can be plucked away and the network will endure. But if just a few of the highly connected nodes are eliminated, the whole system could crash.

Not everyone believes that a universal law is at hand. A recent paper by Boston University physicists found deviations from the power-law pattern in a number of different networks, suggesting a more complicated story. But even so, the study found hidden orders that were far more interesting than the purely random patterns scientists have long used to analyze networks.

''The important point is that the networks are very different from our familiar model systems,'' said Dr. Mark Newman, a mathematician at the Santa Fe Institute. ''This means that all our previous theories have to be thrown out.''

It has only been in recent years that computer power has grown enough to gather and analyze data on such intricate systems. In a highly publicized paper in 1998, Dr. Duncan Watts, a sociologist at Columbia University, and Dr. Steven Strogatz, an applied mathematician at Cornell University, found that many networks exhibited what they called the small-world phenomenon, popularized in John Guare's play ''Six Degrees of Separation.''

Just as any two people can be linked by a chain of no more than about six acquaintances, so can any node in a small-world network be reached from any other node with just a few hops. The two scientists found this hidden order in three networks that could hardly seem more different: the web of neurons forming the simple nervous system of the worm Caenorhabditis elegans, the web of power stations forming the electrical grid of the Western United States and (the finding that attracted the most attention) the web of actors who have appeared together in films.